Optimal Control without Optimization

The real-time barrier in optimal control of nonlinear dynamical systems remains a longstanding limitation across science, engineering, and economics.

Existing approaches rely on iterative optimization and therefore cannot compute optimal control actions directly within physical time for complex, high-dimensional systems.

Here we introduce the Dual Cost-Constraint projection (DCC), a closed-form dynamical representation that enables real-time solutions for a broad class of pseudoconvex optimal control and optimization problems and demonstrates that real-time optimal control can admit a direct closed-form representation.

Unlike classical formulations with Lagrange multipliers or adjoint state variables, the proposed DCC embeds constraints directly within the system dynamics.

The derivation further reveals a structural equivalence between interior-point optimization and nonlinear feedback control, linking constrained optimization with classical stability theory.

Through theoretical analysis and real-world-relevant numerical studies - including autonomous system control, biomechanics monitoring, and economic decision processes - we show that DCC achieves accurate optimal behavior even for highly nonlinear and high-dimensional systems.

The numerical benchmark experiments suggest that DCC controls a 1000-dimensional system at 1 kHz using 77% of a modern CPU, whereas sequential quadratic programming, a widely used state-of-the-art solver for this class of problems, requires more than 80 processors to achieve comparable performance.

Beyond its computational advantages, DCC provides a system-level, causally deterministic interpretation of constrained optimization, revealing that optimal behavior in a broad class of optimal control problems can emerge as the stable evolution of the system itself.

These results lay the foundation for extending this perspective to more general problem classes.